Finite Group Databasebeta

← #49121789

Subgroups — #49121789

Order 36224.

344 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827232C211yesmaximal, characteristic, complemented
2#31826470C212yesmaximal, complemented
3#31826439C211yesmaximal, characteristic, complemented
4#31826221C212yesmaximal, complemented
5#535×C283C211yesmaximal, characteristic
6#22583617—24no—
7C2×#17985376—22no—
8C2×#17985376C2212yes—
9#22583569C2212yes—
10#22583531—24no—
11#22583525—22no—
12C2×#41×C283C2211yescharacteristic
13C22×D4×C283C2211yescharacteristic
14#127×C283C412yescomplemented
15#124×C283C412yescomplemented
16#121×C283—22no—
17#120×C283—22no—
18#117×C283C2211yescharacteristic
19#17985376—24no—
20#17985376—22no—
21#17985376—22no—
22C2×C283⋊C8—24no—
23C2×C283⋊C8D414yescomplemented
24C4⋊Dic283—22no—
25C22⋊Dic283—22no—
26C4×Dic283—22no—
27C24×C283—21no—
28#41×C283—22no—
29#41×C283—22no—
30C2×Q8×C283C2×C411yescharacteristic
31C2×D4×C283—24no—
32C2×D4×C283—22no—
33C2×D4×C283—21no—
34C2×D4×C283—22no—
35C2×D4×C283C2×C412yes—
36C2×D4×C283C2×C411yescharacteristic
37C2×D4×C283D412yes—
38C22×C4×C283D412yes—
39C22×C4×C283C2311yescharacteristic
40C4⋊C4×C283—21no—
41C22⋊C4×C283—21no—
42C22⋊C4×C283—24no—
43C42×C283C2×C412yes—
44C2×Dic283—42no—
45C283⋊C8—24no—
46C23×C283—22no—
47C23×C283—42no—
48C23×C283—82no—
49C23×C283C22⋊C412yes—
50C23×C283C2×D411yescharacteristic
51Q8×C283—22no—
52D4×C283—24no—
53D4×C283—22no—
54D4×C283—24no—
55D4×C283—41no—
56D4×C283—41no—
57D4×C283—84no—
58C2×C4×C283—22no—
59C2×C4×C283—22no—
60C2×C4×C283—22no—
61C2×C4×C283C22⋊C412yes—
62C2×C4×C283C22×C411yescharacteristic
63C2×C4×C283C2×D412yes—
64C2×C4×C283C2×D411yescharacteristic
65C2×C4×C283#4112yes—
66C22×C283—22no—
67C22×C283—41no—
68C22×C283—42no—
69C22×C283—44no—
70C22×C283—42no—
71C22×C283—84no—
72C22×C283C2×C22⋊C411yescharacteristic
73C22×C283#12112yes—
74C4×C283—22no—
75C4×C283—42no—
76C4×C283C4×D412yes—
77C4×C283#12011yescharacteristic
78C4×C283#12311yescharacteristic
79Dic283—82no—
80C2×C283—22no—
81C2×C283—21no—
82C2×C283—42no—
83C2×C283—82no—
84C2×C283#38611yescharacteristic
85C283#191011yescharacteristic, complemented, Sylow
86#1910—2831nomaximal, Sylow
87#568—2831no—
88#535D28311yescharacteristic
89#418—2832no—
90#410—2831no—
91H(3,ℤ/4)—2832no—
92C2×#41C2×D28311yescharacteristic
93C22×D4C2×D28311yescharacteristic
94#131—5664no—
95C2×M16—2832no—
96C2×M16—5662no—
97#127Dic28312yescomplemented
98#124Dic28312yescomplemented
99#121—22no—
100#120—22no—
101#117—2832no—
102#117C2×D28311yescharacteristic
103#102—5664no—
104#99—5662no—
105C24—21no—
106#41—22no—
107#41—22no—
108C2×Q8C2×Dic28311yescharacteristic
109C2×D4—24no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4C2×Dic28312yes—
114C2×D4C2×Dic28311yescharacteristic
115C2×D4C22⋊D28312yes—
116C22×C4C22⋊D28312yes—
117C22×C4C22×D28311yescharacteristic
118M16—5664no—
119M16—5662no—
120M16—5662no—
121C2×C8—2834no—
122C2×C8—5664no—
123C4⋊C4—21no—
124C4⋊C4—5662no—
125C22⋊C4—21no—
126C22⋊C4—24no—
127C22⋊C4—5662no—
128C42—5662no—
129C42C2×Dic28312yes—
130C23—22no—
131C23—42no—
132C23—82no—
133C23C22⋊Dic28312yes—
134C23C2×C22⋊D28311yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—24no—
139D4—41no—
140D4—41no—
141D4—84no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—11322no—
146C2×C4C22⋊Dic28312yes—
147C2×C4C22×Dic28311yescharacteristic
148C2×C4D4×D28312yes—
149C2×C4C2×C22⋊D28311yescharacteristic
150C2×C4#1798539112yes—
151C8—5664no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—44no—
156C22—42no—
157C22—84no—
158C22C2×C22⋊Dic28311yescharacteristic
159C22#2258358212yes—
160C4—22no—
161C4—42no—
162C4—22642no—
163C4D4×Dic28312yes—
164C4#2258357811yescharacteristic
165C4#2258359111yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#3182634511yescharacteristic
subgroupG/H
centerC2#31826345
commutatorC2×C4×C283C22×C4
FrattiniC22×C4C22×D283
Fitting, prime radical#535×C283C2
socleC2×C283#386
cosocleC22×C4×C283C23
hypercenter#535D283
hypocenterC283#1910
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #31827232C2semidirect
2. #31826470C2semidirect
3. #31826439C2semidirect
4. #31826221C2semidirect
14. #127×C283C4semidirect
15. #124×C283C4semidirect
23. C2×C283⋊C8D4semidirect
85. C283#1910semidirect
97. #127Dic283semidirect
98. #124Dic283semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#318272322
#318264702
#318264392
#318262212
#535×C2832
#1910283